Exercises: Arithmetic progressions

Look at the following sequences of numbers. Your task is to identify which of them are arithmetic progressions.

For those that are, state the constant common difference ($d$) between their terms and calculate the number that would go in the next position.

1. a) $5, 9, 13, 17, \dots$
2. b) $2, 4, 8, 16, \dots$
3. c) $20, 15, 10, 5, \dots$
4. d) $1, 1, 2, 3, 5, \dots$

In a supermarket, they are setting up a display of tomato tins in the shape of a flat triangular pyramid against a wall.

In the top row (the top) they place 1 can. In the row just below they place 2 cans, in the third row 3 cans, and so on, increasing by one can for each level they go down.

If the pyramid has a total height of 8 rows:

1. a) Write the sequence that represents the number of cans in each row.
2. b) How many cans will there be in the last row (the base)?

Given the following arithmetic progression:

1. Identify the first term ($a_1$) and the common difference ($d$).

2. Using the general term formula $a_n = a_1 + (n-1) \times d$, calculate the value of the term in position 20 ($a_{20}$).

Ioana has decided to save up to buy a new bicycle. She already has 50 € in her piggy bank from her birthday. She has decided that, starting next week, she will put away 5 € of her pocket money every week without fail.

1. a) Write down the total savings Ioana will have at the end of week 1, week 2, week 3 and week 4.
2. b) What type of progression is it, and what is the common difference?
3. c) Calculate how much money she will have saved in week 12.

Calculate the sum of the first 15 terms of an arithmetic progression, knowing that:

• The first term is $a_1 = 3$.

• The last term to be added is $a_{15} = 45$.

Remember to use the formula for the sum of an arithmetic progression:

In a restaurant, square tables are put together to form a long celebration table.

• If 1 table is used on its own, it seats 4 people (one per side).

• If 2 tables are put together, they seat 6 people.

• If 3 tables are put together, they seat 8 people.

Look at the diagram and answer:

1. a) How many people can be seated if we put 4 tables together?
2. b) What is the constant difference in the number of people that can be seated for each extra table added?
3. c) Calculate how many people could be seated if we put a row of 10 tables together.

Find the general term ($a_n$) of the following arithmetic progressions given by their first terms. Express it in the simplest form possible (for example: $a_n = 3n + 2$).

1. a) $2, 4, 6, 8, \dots$
2. b) $5, 8, 11, 14, \dots$
3. c) $10, 8, 6, 4, \dots$

A bamboo plant grows very quickly and steadily under controlled laboratory conditions.

On day 1, the plant is 15 cm tall.

On day 2, the plant is 18 cm tall.

Knowing that its growth follows an arithmetic progression:

1. a) How many centimetres does the plant grow each day ($d$)?
2. b) How tall will the plant be on day 7?

In a 3rd Year exam, students are asked to calculate the term in position 12 ($a_{12}$) in the decreasing arithmetic progression: $18, 14, 10, 6, \dots$

Aoife solves it in her notebook as follows:

"The difference is $d = 4$ because the numbers go by 4s.

I apply the general term formula: $a_{12} = a_1 + 12 \times d = 18 + 12 \times 4 = 18 + 48 = 66$."

1. a) Find and explain in detail the two conceptual errors made by Aoife in her approach.
2. b) Correct the exercise step by step, calculating the real value of the difference $d$ and the correct value of the term $a_{12}$.
3. c) Without doing complex operations, why is it evident that the result obtained by Aoife ($66$) makes no sense at all looking at the first terms of the sequence?

Imagine you are the one designing the exercises for a written test for your class. You must invent two different arithmetic progressions that satisfy the same condition: in both, the eighth term must be exactly 30 ($a_8 = 30$).

1. a) Construct a first progression that is increasing and whose difference is an even number ($d > 0$). State the value you have chosen for $d$, calculate its first term $a_1$, and write the simplified expression of its general term $a_n$.
2. b) Construct a second progression that is decreasing ($d < 0$) and whose first term is greater than 50 ($a_1 > 50$). Show the values chosen for $d$ and $a_1$, and write its general term $a_n$.
3. c) Check numerically in both progressions that, by substituting $n = 8$ into your formulas, exactly 30 is obtained.

Given the arithmetic sequence formed by the numbers: $7, 11, 15, 19, \dots$

1. a) Write the formula for the general term $a_n$.
2. b) Calculate the value of the term in position 30 ($a_{30}$).
3. c) Determine which position the number $127$ occupies in this sequence.

In an arithmetic progression, we know that the third term ($a_3$) is equal to $12$ and the eighth term ($a_8$) is equal to $27$.

Calculate the common difference $d$ of the progression and write down the value of the first term $a_1$.

Cian is saving up to buy a new bicycle. In the first week, he puts $25$ € into his piggy bank. He has set a goal to increase the amount he saves each week by $2.50$ € compared to the previous week. That is, in the second week he saves $27.50$ €, in the third $30$ €, and so on.

If he continues at this rate:

1. a) How much money will he have to put into the piggy bank in week 12?
2. b) How much money will he have saved in total by the end of week 12?

An auditorium has its seats arranged in the shape of a trapezium. The first row (the closest to the stage) has $15$ seats. Each subsequent row has $2$ more seats than the previous one.

If the auditorium has a total of $20$ rows:

1. a) How many seats does the last row have?
2. b) What is the total capacity (total number of seats) of the auditorium?

Calculate the sum of all the multiples of $3$ between $100$ and $200$.

The measures of the four angles of a quadrilateral form an arithmetic progression. Knowing that the smallest angle measures $60^{\circ}$, calculate the measures of the other three angles.

Remember: The sum of the interior angles of a quadrilateral is always $360^{\circ}$.

In a warehouse, PVC pipes are being stacked to form a triangular structure. In the bottom row (the base), they have placed $18$ pipes. In the row immediately above, they have placed $17$ pipes, in the next $16$, and so on until the top of the stack is reached with a single pipe at the peak.

1. a) How many rows of pipes have been stacked in total?
2. b) How many pipes are there in total in the structure?

We want to place $5$ signposts along a straight road connecting two towns located at kilometre $10$ and kilometre $28$ respectively. The posts must be placed between the two towns in such a way that the distance between consecutive posts is always the same, and is also equal to the distance from the first town to the first post and from the last post to the second town.

Calculate the exact kilometre marker at which each of the $5$ posts should be placed.

To prepare for the Women's Mini Marathon (10 km) in Dublin, Aoife plans a training regimen of 14 consecutive days. On the first day, she runs $3\text{ km}$, and each day she decides to increase the distance covered by $0.5\text{ km}$ compared to the previous day ($a_1 = 3$, $d = 0.5$).

1. a) Estimate: Before calculating anything with exact formulas, estimate mentally whether the total accumulated distance over the 14 days will exceed $80\text{ km}$. To justify your estimate, think about the approximate distance he will run halfway through the plan (for example, on day 7) and multiply it by the 14 days. Explain why this product is a good approximation of the total sum.
2. b) Calculate: Determine the exact distance she will run on the last day ($a_{14}$) and, applying the formula for the sum of an arithmetic progression $S_n = \frac{(a_1 + a_n) \times n}{2}$, calculate the total kilometres she will have covered by the end of the 14 days.
3. c) Check and assess: Compare your initial estimate with the exact result. Was your prediction accurate? What mathematical relationship exists between the value of the middle day and the mean of the first and last terms $\frac{a_1 + a_{14}}{2}$?

According to records from Met Éireann, in a stable atmosphere the temperature decreases at an approximately constant rate of $0.65\,^\circ\text{C}$ for every $100\text{ m}$ of ascent in altitude (vertical lapse rate). On a clear spring day at 11:00, a temperature of $14.2\,^\circ\text{C}$ is recorded in the village of Glencree (Co. Wicklow), located at an altitude of $1500\text{ m}$. Altimetric checkpoints are established every $100\text{ m}$ of ascent towards the summit of Lugnaquilla: checkpoint 1 is at $1500\text{ m}$ ($a_1 = 14.2$), checkpoint 2 at $1600\text{ m}$, checkpoint 3 at $1700\text{ m}$, etc.

1. a) Model the temperature at the checkpoints as an arithmetic progression: identify the value of the difference $d$ and write the formula for the general term $a_n$ as a function of the checkpoint number $n$.
2. b) The mountain rescue hut is located at an altitude of $2500\text{ m}$. What checkpoint number $n$ corresponds to it, and what temperature will be recorded there at that same time?
3. c) On that same slope, at what exact altitude would the freezing temperature ($0\,^\circ\text{C}$) be reached? Reasonably justify whether this elevation coincides exactly with one of the checkpoints of the progression or lies between two of them.

An adventure park wants to install a new zip line. The design of the structure forms a right-angled triangle as shown in the figure. The engineer has determined that the lengths of the three sides of the triangle (the base on the ground, the vertical post, and the zip line cable) must form an arithmetic progression.

Knowing that the common difference of this progression is $6\text{ m}$ (that is, each side is 6 metres longer than the previous one), calculate:

1. The length of each of the three sides.

2. The total length of cable and material required to form the perimeter of the triangle.

(Hint: Remember Pythagoras' theorem $a^2 + b^2 = c^2$ and express the sides in terms of the common difference).

In a plumbing supply warehouse, PVC pipes are stacked in a trapezoidal structure to save space and maintain balance.

In the bottom row (the base) there are 35 pipes. In the top row (the top) there are 5 pipes. Each row has one pipe less than the row immediately below it.

1. How many rows high is the stack?

2. How many PVC pipes are stored in total in this structure?

The interior angles of a convex pentagon (a 5-sided polygon) are in arithmetic progression.

We know that the sum of the interior angles of a polygon is calculated using the formula $(n-2) \times 180^{\circ}$, where $n$ is the number of sides. If the smallest angle of the pentagon measures $72^{\circ}$:

1. Calculate the common difference $d$ of the progression.

2. Determine the measure of the largest angle.

3. Write down the measures of all five angles.

A cyclist is training for a competition and follows a progressive training plan over 14 days.

The plan states that each day she must cover a specific distance, increasing this distance by a fixed number of kilometres compared to the previous day.

We know that:

• On the third day ($a_3$) she covered 24 km.

• On the last day ($a_{14}$) she covered 79 km.

Answer with explanations:

1. How many kilometres did she cover on the first day?

2. What is the daily increase in distance ($d$)?

3. Calculate the total accumulated distance she will have covered by the end of the 14 days of training.

Three numbers form an arithmetic progression. We know that their sum is equal to 33 and their product is equal to 1287.

Set up a system of equations using the properties of arithmetic progressions (you can call the terms $x-d$, $x$ and $x+d$ to simplify) and find the values of these three numbers.

The seating area of a music auditorium has the shape of an isosceles trapezoid. Due to this shape, each row has more seats than the previous one.

• The first row (the closest to the stage) has 20 seats.

• The auditorium has a total of 780 seats.

• Each row has 2 more seats than the row immediately before it.

Using the sum formulas for an arithmetic progression ($S_n$), set up a quadratic equation to find out how many rows ($n$) the auditorium has in total.

A safety fence needs to be installed along a straight road. The fence posts must be placed at regular intervals forming an arithmetic progression.

We know that:

• Post number 11 is located at kilometre marker 5.4.

• The distance between post number 5 and post number 20 is 450 metres ($0.45\text{ km}$).

Calculate:

1. The distance between each post (common difference of the progression).

2. At which kilometre marker the first post will be placed.

A tech company offers two types of salary contracts to an engineer for a 5-year project (60 months):

• Option A: An initial monthly salary of $\text{€}\ 1500$, with an increase of $\text{€}\ 20$ each month compared to the previous month (monthly arithmetic progression).

• Option B: A fixed salary of $\text{€}\ 24,000$ in the first year, with an annual increase of $\text{€}\ 3,000$ each subsequent year (annual arithmetic progression).

Calculate the total money earned over the full 5 years with each option and determine which is the most economically advantageous for the engineer. (Note: In Option A you must sum 60 terms; in Option B you must sum 5 annual terms).

In an arithmetic progression, we know neither the first term nor the difference, but we know that the sixth term is 19 ($a_6 = 19$) and the fifteenth term is 64 ($a_{15} = 64$). We need to find the term in position 40 ($a_{40}$).

Two 3rd Year students propose two different strategies:

• Seán's method: Set up the system of two equations with two unknowns formed by $a_1 + 5d = 19$ and $a_1 + 14d = 64$, solve it to find $a_1$ and $d$, and then calculate $a_{40} = a_1 + 39d$.

• Aoife's method: Notice that moving from term 6 to term 15 takes 9 steps: $a_{15} = a_6 + 9d$. Solve for $d$ directly from that single equation and then calculate $a_{40}$ moving forward from $a_{15}$ using the formula $a_{40} = a_{15} + (40 - 15) \times d$, without calculating $a_1$.

1. a) Solve the problem following the steps of Seán's method until you find $a_{40}$.
2. b) Solve the problem following the steps of Aoife's method and check whether you arrive at the same solution.
3. c) Compare both methods: which one do you consider more efficient and with less risk of making calculation errors? Justify your answer.
4. d) If the problem also asked you to calculate the sum of the first 40 terms ($S_{40}$), would Aoife's method still be shorter, or would Seán's approach gain an advantage? Reason why.

An urban parcel delivery company in Dublin renews its fleet with electric vans under a 24-month (2-year) leasing contract. In the first month of use, a van travels $3200\text{ km}$ ($a_1 = 3200$). Due to gradual battery degradation and maintenance, the company assumes that each month the van will travel a fixed number of kilometres less than the previous month, describing a decreasing arithmetic progression with difference $d$.

1. a) The company establishes that, for logistical profitability, the mileage of the last month of the contract (month 24) must not be less than $2050\text{ km}$ ($a_{24} \ge 2050$). Determine the maximum allowable monthly reduction (the limit value of the difference $d$).
2. b) Assuming that the monthly reduction is exactly that limit difference found in the previous section, calculate the total accumulated distance that the van will have covered during the 24 months of the contract ($S_{24}$).
3. c) In a management meeting, a manager proposes an alternative commercial objective: "Keeping the initial $3200\text{ km}$ in month 1, we must ensure that the van accumulates a total of at least $80\,000\text{ km}$ over the 24 months, allowing the mileage to decrease month by month ($d < 0$)". Prove mathematically, using the properties of the sum of an arithmetic progression, whether this manager's proposal is possible or mathematically unfeasible.